The Distribution of p-Torsion in Degree p Cyclic Fields

TL;DR

This paper computes the moments of p-torsion in class groups of cyclic degree p fields, confirming a distribution conjecture extending Cohen-Lenstra-Martinet, with unconditional results for p=3 and conditional under GRH.

Contribution

It provides the first unconditional computation for p=3 and extends the distribution conjecture to all primes p under GRH, following methods similar to Fouvry and Kluners.

Findings

Unconditional moments computed for p=3.

Distribution matches Gerth's conjecture.

Extension of Cohen-Lenstra-Martinet conjectures.

Abstract

We compute all the moments of the p-torsion in the first step of a filtration of the class group defined by Gerth for cyclic fields of degree p, unconditionally for p=3 and under GRH in general. We show that it satisfies a distribution which Gerth conjectured as an extension of the Cohen-Lenstra-Martinet conjectures. In the p=3 case this gives the distribution of the 3-torsion of the class group modulo the Galois invariant part. We follow the strategy used by Fouvry and Kluners in their proof of the distribution of the 4-torsion in quadratic fields.